The Limit of a Function (Part 1)

These flashcards cover the fundamental concepts of limits in calculus, focusing on definitions and key terms related to limits as discussed in the lecture.

Limit of a Function

A fundamental concept in calculus that describes the behavior of a function as the input approaches a certain value.

lim x→a f(x) = L

Notation that indicates the limit of f(x) as x approaches a equals L.

Left-Hand Limit

The limit of a function as x approaches a from the left, denoted as lim x→a− f(x).

Right-Hand Limit

The limit of a function as x approaches a from the right, denoted as lim x→a+ f(x).

One-Sided Limits

Limits that consider the behavior of a function approaching a point from one side only.

The limit exists

If lim x→a f(x) = L if and only if lim x→a− f(x) = L and lim x→a+ f(x) = L.

Continuous Function

A function where the limit as x approaches a value equals the actual value of the function at that point.

Discontinuous Function

A function where the limit as x approaches a value does not equal the actual value of the function at that point.

Limit does not exist

Occurs when the left-hand limit and right-hand limit at a point are different.

Behavior of f(x) approaching a value

The values of f(x) get closer to a limit L as x approaches a from either side.

Sufficiently close to a

Refers to how close x must be to value a (not equal to a) to analyze limits.